What is Trigonometric: Definition and 1000 Discussions

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.Trigonometry is known for its many identities. These
trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.

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  1. V

    Understanding Trigonometric Functions and Their Geometric Meaning

    I'm not sure if this is the correct section for this thread since this isn't homework, but my question is very basic, so I think this section is suitable. I have two questions regarding the trigonometric functions (sinx,cosx,tanx etc). 1) What is the geometric meaning (i.e in the context...
  2. anemone

    MHB Trigonometric Challenge V

    Prove that $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)=\tan 9^{\circ}$
  3. anemone

    MHB Can CSI and AM-GM Inequalities Solve Trigonometric Equation?

    Solve the equation $\sin a \cos b+ \sin b \cos c+ \sin c \cos a=\dfrac{3}{2}$
  4. S

    A tricky trigonometric problem

    Homework Statement A cubic equation is given as: ##x^{3} -(1+\cos \theta +\sin \theta)x^{2} +(\cos \theta \sin \theta +\cos \theta +\sin \theta)x-\sin \theta \cos \theta=0## Show that x=1 is a root of the equation for all values of θ and deduce that x-1 is a factor to the above equation...
  5. E

    Help with Trigonometric Integrals

    Could someone help me with these two problems? I've been at them for an hour, but have very little clue how to go about solving either of them. Homework Statement 1)∫ 6 csc^3 (x) cot x dx Homework Equations The Attempt at a Solution 6 ∫ csc^3 (x) dx) / tan x csc^3 / tan x =...
  6. anemone

    MHB Can You Prove That $\tan 50^{\circ}>1.18$ Without a Calculator?

    Without the help of calculator, show that $\tan 50^{\circ}>1.18$
  7. anemone

    MHB Trigonometric Challenge II

    Let $k$ be an odd positive integer. Solve the equation $\cos kx=2^{k-1} \cos x$.
  8. S

    How do I find the remaining solutions for the trigonometric equation?

    Homework Statement 4sin2(2x) -1 = 0 Solve over 0 --> 2pi The attempt at a solution (2sin(2x) + 1) (2sin(2x) - 1) = 0 2sin(2x) +1 = 0 sin(2x) = -1/2 2x = -pi/6 x = -pi/12 and -11pi/12 which = 23pi/12 and 13pi/12. 2sin(2x) - 1 = 0 sin(2x) = 1/2 2x = pi/6 x= pi/12 and...
  9. anemone

    MHB Can you prove this trigonometric equation? 3cos(p+s)=7cos(q+r)

    Let $p,\,q,\,r,\,s\,\in[0,\,\pi]$ and we are given that $2\cos p+6 \cos q+7 \cos r+9 \cos s=0$ and $2\sin p-6 \sin q+7 \sin r-9 \sin s=0$. Prove that $3 \cos (p+s)=7\cos(q+r)$.
  10. MarkFL

    MHB Trentan's question at Yahoo Answers regarding a trigonometric equation

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  11. anemone

    MHB Show the trigonometric equation has no solutions

    Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.
  12. T

    Trigonometric and hyper. functions approx in large small argumnt

    hello guys , i'm looking for approximation of trigonometric and hyperbolic functions for small and large argument, is it correct to say sin(x)=x and tg(x)=x and tgh(x)=x and cos(x) = 1 and cosh(x)=1 and coth(x)=1/x for small x what about large x ? what can we say about exponential function...
  13. Albert1

    MHB Another trigonometric equality

    the units of all angles :degree gien :$tan\, x\,+tan\,(x+60)\,-\,tan(60-x)=3tan(3x)$ prove :$tan^2\, x\,+tan^2\,(x+60)\,+\,tan^2(60-x)=9tan^2(3x)+6$
  14. Albert1

    MHB Prove Trigonometric Equality: $tan 1^o+tan 5^o+tan 9^o = tan 177^o-45$

    prove: $tan 1^o+tan 5^o+tan 9^o +---------+tan 177^o=45$
  15. Albert1

    MHB What values of tan alpha and tan beta satisfy a trigonometric inequality?

    $0<\alpha < \dfrac {\pi}{2}$ $0<\beta < \dfrac {\pi}{2}$ prove: $(1): \,\, \dfrac{1}{ \cos^2 \alpha}+ \dfrac {1}{ \sin^2 \alpha \, \sin^2 \beta \, \cos^2 \beta} \geq 9 $ determine the values of $ \tan \alpha$ and $ \tan \beta $ when : $(2): \: \dfrac{1}{ \cos^2 \alpha}+ \dfrac {1}{ \sin^2 \alpha...
  16. A

    MHB Reduction formula instead of using identities for trigonometric integration?

    This is one of the example problems in my book to show how to deal with integrating trigonometric functions to higher powers, by breaking them down into identities. =\int cos^5x dx =\int (cos^2x)^2cos^x dx =\int (1-sin^2x)^2*d(sin x) =\int (1-u^2)^2 du =\int 1-2u^2 + u^4 du =u-\frac{2}{3}u^3...
  17. karush

    MHB Can We Use Tan^2 Theta to Solve Trig Substitutions?

    $\displaystyle \int {\frac{\sqrt{x^2-9}}{x}}\ dx $ using $\displaystyle x=3\sec{\theta}\ \ \ dx=3\sin{\theta}\sec^2{\theta}\ d\theta $ so then $\displaystyle \int {\frac{3\tan{\theta}}{3\sec{\theta}}}\ 3\sin{\theta}\sec^2{\theta}\ d\theta \Rightarrow 3\int {\tan^2{\theta}}\ d\theta $ the...
  18. anemone

    MHB Solving exponential (of trigonometric functions) equation

    Hi MHB, Solve $(2+ \sqrt{2})^{(\sin x)^2}-(2- \sqrt{2})^{(\cos x)^2}=\left( 1+ \dfrac{1}{\sqrt{2}} \right)^{\cos 2x} -(2-\sqrt{2})^{\cos 2x}$. This problem vexes me much because the only way that I could think of to solve this problem would be by substituting $(\sin x)^2=u$, and from there, I...
  19. anemone

    MHB Solve a trigonometric equation

    Let $y$ be in radians and $0<y<\dfrac{\pi}{4}$. Solve for $y $ if $\tan 4y=\dfrac{\cos y-\sin y}{\cos y +\sin y}$.
  20. W

    Indefinite trigonometric integral with an Nth Root

    Homework Statement Solve: \int sin(16x) \sqrt[a]{cos(16x)}\,dx Answer should be linear in the constant "a" The Attempt at a Solution \int sin(16x) \sqrt[a]{cos(16x)}\,dx Set: u=cos(16x), du=-16sin(16x) du ~~\Rightarrow~~ {-1/16}\int \sqrt[a]{u}\,du =...
  21. S

    MHB Can the derivative of the given integral be simplified to -A?

    Can it be proved? \left(\frac{-2\sin A}{1-\cos A}\right)\cos\left(\frac{A}{2}\right)\tan^{-1}\left[\cos \left(\frac{A}{2}\right)\right]=\frac{\pi^2-4A^2}{8}
  22. J

    Parity of inverse trigonometric functions

    When I place the trigonometric functions in the "wolfram google", it informs the parity of the function, so, sin(x), sinh(x) -> odd cos(x), cosh(x) -> even tan(x), tanh(x) -> odd cot(x), coth(x) -> odd sec(x), sech(x) -> even csc(x), csch(x) -> odd arcsin(x), arcsinh(x) -> odd...
  23. anemone

    MHB Evaluate trigonometric expression

    Without the help of calculator, evaluate $\cos \dfrac{\pi}{7}\cos \dfrac{2\pi}{7}\cos \dfrac{4\pi}{7}$.
  24. A

    MHB Integrating Fifth Power of Secant Using Partial Fractions?

    how do i integrate the fifth power of a secant? i broke it up into powers of two and three but that didn't seem to work
  25. A

    MHB Trigonometric Integration

    how do i integrate sin(pi*x)*sqrt(1 + pi*2*cos(pi*x)^2)? i reduced this to sqrt(u^2-u) but i don't know how to go from here
  26. MarkFL

    MHB Integral of sqrt(1+x^2)/x: Help Solving w/ Trig Substitution

    Here is the question: I have posted a link there to this question so the OP can view my work.
  27. S

    Proving trigonometric identities in converse

    Homework Statement If ##\sec x-\csc x=\pm p##, show that ##p^{2} \sin^2 2x +4\sin 2x-4=0## Show conversely that if ##p^{2} \sin^2 2x +4\sin 2x-4=0##, then ##\sec x-\csc x## is equal to +p and -p. Find, to the nearest minute, the two values of x in the range of 0 to 360 degrees, the equation...
  28. talknerdy2me

    Rearranging Trigonometric Functions

    This is my very first post - so i hope I don't break any rules - its more of a formula rearranging question/confirmation so here goes... Homework Statement Currently working on friction - static/kinetic - so in my textbook it states in a side bar "info bit" that tanθ=sinθ/cosθ my...
  29. E

    Approximating function by trigonometric polynomial

    Hi! Say that we wish to approximate a function f(x), \, x\in [0, 2\pi] by a trigonometric polynomial such that f(x) \approx \sum_{|n|\leq N} a_n e^{inx} \qquad (1) The best approximation theorem says that in a function space equipped with the inner product (f,g) = \frac{1}{2...
  30. S

    MHB Trigonometric Integrals [1]

    Stuck on this problem. Evaluate \int \cos^{2}x \, \tan^{3}x \, dx What I have so far: used the trig identity sin/cos = tan factored out a sin so I can have a even power. changed \sin^{2}x to its identity = 1/2(1 - cos2x) combined like terms and canceled out the cos \int \cos^{2}x *...
  31. S

    Help finding integral of trigonometric function

    Homework Statement 4∫tan(x^2)dx from 0 to √(π)/2Homework Equations 4∫tan(x^2)dx from 0 to √(π)/2The Attempt at a Solution I tried doing u-substitution, which didn't work, and also tried to look for a trig identity and wasn't able to find any relevant one.
  32. S

    MHB Can I Simplify Trigonometric Integrals by Taking out Constants?

    Quick question. \int sin^{4}x dx so I know: \frac{1}{2} \int 1 - 2cos2x + \frac{1}{2}(1 + cos4x)dx So here I first brought out the 1/2 because it's a constant and it's nasty. so now I have \frac{1}{4} \int 1 - 2cos2x + 1 + cos4x dx so...Just as I brought 1/2 out can I now precede to take...
  33. anemone

    MHB Sum of two trigonometric terms

    Prove that $\tan \left( \dfrac{3 \pi}{11} \right)+ 4\sin \left( \dfrac{2 \pi}{11} \right)=\sqrt{11}$. I know this problem may be stale as it has been posted countless times at other math forums, but I've seen one brilliant method to attack this problem recently, and I'm so eager to share it...
  34. MarkFL

    MHB Verify Trig Identity: 1+cosx+cos2x=1/2+(sin5/2x)/(2sin1/2x) - Catlover0330

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  35. K

    MHB Trigonometric inequality bounded by lines

    How can it be shown that $$16x\cos(8x)+4x\sin(8x)-2\sin(8x)<|17x|?$$ This problem arises from work with damped motion in spring-mass systems in Differential Equations. I have gotten to this inequality after some algebraic manipulation, but am completely stuck here. Here is the illustrative...
  36. anemone

    MHB Proving a Trigonometric equality

    Hi MHB, I have found this problem quite interesting to me and hence I have spent some time on it but all of my attempts to prove it went down the drain. I have no choice but posting it here, hoping to gain some insight from the members of the forum on how to prove this problem. Thanks in...
  37. D

    Trigonometric interpolation polynomial

    Homework Statement Let t_j=j/100, a_j=j, b_j=-j, for j=0,1,...,99. Define f(t)=\sum\limits_{k=0}^{99} (a_k\cos(2\pi kt)+b_k\sin(2\pi kt)) Determine the values of c_l, d_m for l= 0,...5, m=1,...,4, so that P(t)=\frac{c_0}{2}+\sum\limits_{k=1}^4 (c_k\cos(2\pi kt)+d_k\sin(2\pi kt))+c_5\cos(10\pi...
  38. K

    Definite integration of Trigonometric Functions

    Homework Statement [0,1]∫(3x)dx/(4-3x)^1/2 (3xdx divided by square root of 4-3x) Homework Equations The Attempt at a Solution I could not get the bookish answer of that...actually my answer was wholly different... i let 4-3x (without square root) = t and then use substitution...
  39. J

    Derivative and trigonometric functions

    Hellow! If we can equal the first derivative with a trigonometric function: \frac{dy}{dx}=tan(\theta) So, the second derivative is equal to which trigonometric function? \frac{d^2y}{dx^2}=? Thanks!
  40. DreamWeaver

    MHB Trigonometric series related to the Hurwitz Zeta function

    This thread is dedicated to exploring the trigonometric series shown below. This is NOT a tutorial, so all and any contributions would be very much welcome... (Heidy)\mathscr{S}_{\infty}(z)= \sum_{k=1}^{\infty}\frac{\log k}{k^2}\cos(2\pi kz) This series can be expressed in terms of the...
  41. S

    MHB Trigonometric Identity Questions

    Your help will be greatly appreciated! Thanks!1. The expression \(\sin\pi\) is equal to \(0\), while the expression $\frac{1}{\csc\pi}$ is undefined. Why is $\sin\theta=\frac{1}{\csc\theta}$ still an identity? 2. Prove $\cos(\theta + \frac{\pi}{2})= -\sin\theta$
  42. anemone

    MHB Trigonometric Inequality Challenge

    For any triangle $ABC$, prove that $\cos \dfrac{A}{2} \cot \dfrac{A}{2}+\cos \dfrac{B}{2} \cot \dfrac{B}{2}+\cos \dfrac{C}{2} \cot \dfrac{C}{2} \ge \dfrac{\sqrt{3}}{2} \left( \cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot \dfrac{C}{2} \right)$
  43. anemone

    MHB How can I express this trigonometric equation using cosine of 3x?

    Express $\cos^7 x+\cos^7 \left( x+\dfrac{2 \pi}{3} \right)+\cos^7 \left( x+\dfrac{4 \pi}{3} \right)$ in terms of $\cos 3x$.
  44. K

    Integration problem in trigonometric functions

    Homework Statement please help me with this integration problem? ∫(1/sinx+ cosx) dx Homework Equations i don't know any proper substitution in this question,maybe there are none The Attempt at a Solution i tried rationalizing and it has got me this far...
  45. karush

    MHB Trigonometric equation: 2cos(θ) + 2sin(θ) = √(6)

    $2\cos{\theta}+2\sin{\theta}=\sqrt{6}$ $\displaystyle\cos{\theta}+\sin{\theta}=\frac{ \sqrt{6} }{2}$ $\displaystyle(\cos{\theta}+\sin{\theta})^2=\frac{3}{2}$ $\displaystyle\cos^2{\theta}+2cos\theta\sin\theta+\sin{\theta}^2=\frac{3}{2}$ $\displaystyle\sin{2\theta}=\frac{1}{2} \Rightarrow...
  46. T

    Estimating the integral of a decreasing trigonometric function

    Hi, So this is part of an assignment for my numerical analysis class. The integral is this: \int_0^{\infty} e^{-x} \cos^2 (x^2) dx We are instructed to evaluate the integral from 0 to some large A using numerical methods (which I'm fine with), and then estimate the tail, ie...
  47. Seydlitz

    Finding trigonometric solution to a cubic equation using computer

    Hello, Homework Statement I get this question from Mathematical Methods by Boas page 74 problem 25. The question states: "Use a computer to find the three solutions of the equation ##x^3-3x-1=0##. Find a way to show that the solutions can be written as ##2cos(\frac{\pi}{9})##...
  48. MarkFL

    MHB Derivative of Inverse Trig Function: y=4*arcsin(x/4)

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  49. M

    Trigonometric Integration: Finding the Solution for ∫√(π) xsin(x²-1) dx

    Homework Statement 0∫\sqrt{∏} xsin(##x^2## -1) dx Not sure how I should be formatting this, but the square root of pi is 'on top of' the integral, and zero is 'below'. The expression to integrate is \sqrt{∏} xsin(##x^2## -1) dx. The Attempt at a Solution As sin integrated is -cos...
  50. F

    Finding Phase Shift in Trigonometric Equations

    Hello, Probably a simple problem, but I am not able to figure it out. a \cos (\epsilon) - b \sin (\epsilon) = c in-phase part a \sin (\epsilon) - b \cos (\epsilon) = d out-of-phase part In order to find the phase shift, the in-phase term has to be divided by the out-of-phase...
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